Last update: 26 February 2013

A. Young invented the theory of standard Young tableaux in order to describe the representation theory of the symmetric group ${S}_{n};$ the group of $n\times n$ matrices such that

- the entries are either 0 or 1,
- there is exactly one nonzero entry in each row and each column.

Young himself began to generalize the theory and in [You1929] he provided a theory for the Weyl groups of type B, i.e. the hyperoctahedral groups $W{B}_{n}\cong (\mathbb{Z}/2\mathbb{Z})\wr {S}_{n}$ of $n\times n$ matrices such that

- the entries are either 0 or $\pm 1,$
- there is exactly one nonzero entry in each row and each column.

In the same paper Young also treated the Weyl group $W{D}_{n}$ of $n\times n$ matrices such that

- the entries are either 0 or $\pm 1,$
- there is exactly one nonzero entry in each row and each column,
- the product of the nonzero entries is 1.

W. Specht [Spe1932] generalized the theory to cover the complex reflection groups ${G}_{r,1,n}\cong (\mathbb{Z}/r\mathbb{Z})\wr {S}_{n}$ consisting of $n\times n$ matrices such that

- the entries are either 0 or $r\text{th}$ roots of unity,
- there is exactly one nonzero entry in each row and each column.

In the classification [STo1954] of finite groups generated by complex reflections there is a single infinite family of groups $G(r,p,n)$ and exactly 34 others, the “exceptional” complex reflection groups. The groups $G(r,p,n)$ are the groups of $n\times n$ matrices such that

- the entries are either 0 or $r\text{th}$ roots of unity,
- there is exactly one nonzero entry in each row and each column,
- the $(r/p)\text{th}$ power of the product of the nonzero entries is 1.

Though we do not know of an early reference which generalizes the theory of Young tableaux to these groups, it is not difficult to see that the method that Young uses for the Weyl groups $W{D}_{n}$ extends easily to handle the groups $G(r,p,n)\text{.}$

Special cases of the groups $G(r,p,n)$ are

- $G(1,1,n)={S}_{n},$ the symmetric group,
- $G(2,1,n)=W{B}_{n},$ the hyperoctahedral group (i.e. the Weyl group of type ${B}_{n}\text{),}$
- $G(2,2,n)=W{D}_{n},$ the Weyl group of type ${D}_{n},$
- ${G}_{r,1,n}\cong (\mathbb{Z}/r\mathbb{Z})\wr {S}_{n}={(\mathbb{Z}/r\mathbb{Z})}^{n}\u22ca{S}_{n}\text{.}$

The order of $G(r,1,n)$ is ${r}^{n}n!\text{.}$ Let ${E}_{ij}$ be the $n\times n$ matrix with 1 in the $(i,j)$ position and all other entries 0. Then $G(r,1,n)$ can be presented by generators

$${s}_{1}=\zeta {E}_{11}+\sum _{j\ne 1}{E}_{ii},\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{s}_{i}={E}_{i,i+1}+{E}_{i+1,i}+\sum _{j\ne i,i+1}{E}_{jj},\phantom{\rule{2em}{0ex}}2\le i\le n,$$where $\zeta $ is a primitive $r\text{th}$ root of unity, and relations

$$\begin{array}{ccc}\text{(B1)}& {s}_{i}{s}_{j}={s}_{j}{s}_{i},& \text{if}\hspace{0.17em}\mid i-j\mid >1,\\ \text{(B2)}& {s}_{i}{s}_{i+1}{s}_{i}={s}_{i+1}{s}_{i}{s}_{i+1},& \text{for}\hspace{0.17em}2\le i\le n-1,\\ \text{(BB)}& {s}_{1}{s}_{2}{s}_{1}{s}_{2}={s}_{2}{s}_{1}{s}_{2}{s}_{1},\\ \text{(C)}& {s}_{1}^{r}=1,\\ \text{(R)}& {s}_{i}^{2}=1,& \text{for}\hspace{0.17em}2\le i\le n\text{.}\end{array}$$The group $G(r,p,n)$ is the subgroup of index $p$ in $G(r,1,n)$ generated by

$${a}_{0}={s}_{1}^{p},\phantom{\rule{2em}{0ex}}{a}_{1}={s}_{1}{s}_{2}{s}_{1},\phantom{\rule{2em}{0ex}}{a}_{i}={s}_{i},\phantom{\rule{1em}{0ex}}2\le i\le n\text{.}$$More recently there has been an interest in Iwahori-Hecke algebras associated to reflection groups and there has been significant work generalizing the constructions of A. Young to these algebras. Iwahori-Hecke algebras of types A, B and D were handled by Hoefsmit [Hoe1974] and other aspects of the theory for these algebras were developed by Dipper, James and Murphy [DJa1987], [DJM1995], Gyoja [Gyo1986] and Wenzl [Wen1988]. In 1994, Ariki and Koike [AKo1994] introduced cyclotomic Hecke algebras ${H}_{r,1,n}$ for the complex reflection groups $G(r,1,n)$ and they generalized the Young tableau theory to these algebras. Theorem 3.18 below shows that the theory of [AKo1994] is a special case of an even more general theory for affine Hecke algebras.

Let ${u}_{1},\dots ,{u}_{r},q\in \u2102,$ $q\ne 0\text{.}$ The cyclotomic Hecke algebra is the algebra ${H}_{r,1,n}({u}_{1},\dots ,{u}_{r};q),$ over $\u2102,$ given by generators ${T}_{1},\dots ,{T}_{n}$ and relations

$$\begin{array}{ccc}\text{(B1)}& {T}_{i}{T}_{j}={T}_{j}{T}_{i},& \text{if}\hspace{0.17em}\mid i-j\mid >1,\\ \text{(B2)}& {T}_{i}{T}_{i+1}{T}_{i}={T}_{i+1}{T}_{i}{T}_{i+1},& \text{for}\hspace{0.17em}2\le i\le n-1,\\ \text{(BB)}& {T}_{1}{T}_{2}{T}_{1}{T}_{2}={T}_{2}{T}_{1}{T}_{2}{T}_{1},\\ \text{(qC)}& ({T}_{1}-{u}_{1})({T}_{1}-{u}_{2})\dots ({T}_{1}-{u}_{r})=0\text{.}\\ \text{(1R)}& ({T}_{i}-q)({T}_{i}+{q}^{-1})=0,& \text{for}\hspace{0.17em}2\le i\le n\text{.}\end{array}$$The algebra ${H}_{r,1,n}({u}_{1},\dots ,{u}_{r};q)$ is of dimension ${r}^{n}n!$ (see [AKo1994]).

- ${H}_{1,1,n}(1;q)$ is the Iwahori-Hecke algebra of type ${A}_{n-1}\text{.}$
- ${H}_{2,1,n}(q,-{q}^{-1};q)$ is the Iwahori-Hecke algebra of type ${B}_{n}\text{.}$
- If $\zeta $ is a primitive $r\text{th}$ root of 1 then ${H}_{r,1,n}(1,\zeta ,\dots ,{\zeta}^{r-1};1)$ is the group algebra $\u2102G(r,1,n)\text{.}$

Fact (c) says that the representation theory of the groups $G(r,1,n)$ is a special case of the representation theory of the algebras ${H}_{r,1,n}\text{.}$

The work of Broué, Malle and Michel [BMM1993] demonstrated that there are cyclotomic Hecke algebras associated to most complex reflection groups (even exceptional complex reflection groups). In particular, there are cyclotomic Hecke algebras ${H}_{r,p,n}$ corresponding to all the groups $G(r,p,n)$ and Ariki [Ari1995] has generalized the Young tableau mechanism to these groups (see also [HRa1998]). Theorems 3.15 and the mechanism of Theorem 2.8 show that the theory of Ariki is a special case of a general construction for affine Hecke algebras.

Let $r,p,n\in {\mathbb{Z}}_{>0}$ be such that $p$ divides $r$ and let $d=r/p\text{.}$ Let ${x}_{0},\dots ,{x}_{d-1}\in \u2102$ and let $\xi $ be a primitive $p\text{th}$ root of 1. For $1\le j\le r,$ define

$${u}_{j}={\xi}^{k}{x}_{\ell},\phantom{\rule{2em}{0ex}}\text{if}\hspace{0.17em}j-1=\ell p+k,\phantom{\rule{2em}{0ex}}(0\le k\le p-1,0\le \ell \le d-1)$$i.e., ${u}_{1},\dots ,{u}_{r}\in \u2102$ are chosen so that

$$({T}_{1}-{u}_{1})({T}_{1}-{u}_{2})\dots ({T}_{1}-{u}_{r})=({T}_{1}^{p}-{x}_{0}^{p})({T}_{1}^{p}-{x}_{1}^{p})\dots ({T}_{1}^{p}-{x}_{d-1}^{p})\text{.}$$The algebra ${H}_{r,p,n}({x}_{0},\dots ,{x}_{d-1};q)$ is the subalgebra of ${H}_{r,1,n}({u}_{1},\dots ,{u}_{r};q)$ generated by the elements

$${a}_{0}={T}_{1}^{p},\phantom{\rule{2em}{0ex}}{a}_{1}={T}_{1}^{-1}{T}_{2}{T}_{1},\phantom{\rule{2em}{0ex}}{a}_{i}={T}_{i},\phantom{\rule{1em}{0ex}}\text{for}\hspace{0.17em}2\le i\le n\text{.}$$Then

- ${H}_{2,2,n}(1;q)$ is the Iwahori-Hecke algebra of type ${D}_{n},$
- If $\eta $ is a primitive $d\text{th}$ root of unity then ${H}_{r,p,n}(1,\eta ,\dots ,{\eta}^{d-1};1)$ is the group algebra $\u2102G(r,p,n)\text{.}$

There are three common ways of depicting affine braids [Cri1997], [GLa1997], [Jon1994]:

- As braids in a (slightly thickened) cylinder,
- As braids in a (slightly thickened) annulus,
- As braids with a flagpole.

See Figure 1. The multiplication is by placing one cylinder on top of another, placing one annulus inside another, or placing one flagpole braid on top of another. These are equivalent formulations: an annulus can be made into a cylinder by turning up the edges, and a cylindrical braid can be made into a flagpole braid by putting a flagpole down the middle of the cylinder and pushing the pole over to the left so that the strings begin and end to its right.

The group formed by the affine braids with $n$ strands is the affine braid group of type A. The affine braid group ${\mathcal{B}}_{\infty ,1,n}$ is presented by generators ${T}_{2},\dots ,{T}_{n}$ and ${X}^{{\epsilon}_{1}}$ (see Figure 2) with relations

$$\begin{array}{ccc}\text{(B1)}& {T}_{i}{T}_{j}={T}_{j}{T}_{i},& \text{if}\hspace{0.17em}\mid i-j\mid >1,\\ \text{(B2)}& {T}_{i}{T}_{i+1}{T}_{i}={T}_{i+1}{T}_{i}{T}_{i+1},& \text{for}\hspace{0.17em}2\le i\le n-1,\\ \text{(BB)}& {X}^{{\epsilon}_{1}}{T}_{2}{X}^{{\epsilon}_{1}}{T}_{2}={T}_{2}{X}^{{\epsilon}_{1}}{T}_{2}{X}^{{\epsilon}_{1}},\\ \text{(B1}\prime \text{)}& {X}^{{\epsilon}_{1}}{T}_{i}={T}_{i}{X}^{{\epsilon}_{1}},& \text{for}\hspace{0.17em}3\le i\le n\text{.}\end{array}$$Inductively define ${X}^{{\epsilon}_{i}}\in {\mathcal{B}}_{\infty ,1,n}$ by

$$\begin{array}{cc}{X}^{{\epsilon}_{i}}={T}_{i}{X}^{{\epsilon}_{i-1}}{T}_{i},\phantom{\rule{2em}{0ex}}2\le i\le n\text{.}& \text{(1.4)}\end{array}$$By drawing pictures of the corresponding affine braids one can check that the ${X}^{{\epsilon}_{i}}$ all commute with each other. View the symbols ${\epsilon}_{i}$ as a basis of ${\mathbb{R}}^{n}$ so that

$$\begin{array}{cc}{\mathbb{R}}^{n}=\sum _{i=1}^{n}\mathbb{R}{\epsilon}_{i},\phantom{\rule{2em}{0ex}}\text{and let}\phantom{\rule{2em}{0ex}}L=\sum _{i=1}^{n}\mathbb{Z}{\epsilon}_{i}\text{.}& \text{(1.5)}\end{array}$$The affine braid group ${\mathcal{B}}_{\infty ,1,n}$ contains a large abelian subgroup

$$\begin{array}{cc}X=\{{X}^{\lambda}\hspace{0.17em}\mid \hspace{0.17em}\lambda \in L\},& \text{(1.6)}\end{array}$$
where
${X}^{\lambda}={\left({X}^{{\epsilon}_{1}}\right)}^{{\lambda}_{1}}\dots {\left({X}^{{\epsilon}_{n}}\right)}^{{\lambda}_{n}}$
for
$\lambda ={\lambda}_{1}{\epsilon}_{1}+\dots +{\lambda}_{n}{\epsilon}_{n}\in L\text{.}$
The *pole winding number* of an affine braid $b\in {\mathcal{B}}_{\infty ,1,n}$
is $\kappa \left(b\right)$ where
$\kappa :{\mathcal{B}}_{\infty ,1,n}\to \mathbb{Z}$
is the group homomorphism defined by
$\kappa \left({X}^{{\epsilon}_{1}}\right)=1$ and
$\kappa \left({T}_{i}\right)=0,\hspace{0.17em}2\le i\le n\text{.}$
The affine braid group ${\mathcal{B}}_{\infty ,p,n}$ is the subgroup of
${\mathcal{B}}_{\infty ,1,n}$ of affine braids with pole winding number equal to 0
(mod $p\text{),}$

Define

$$\begin{array}{cc}Q=\sum _{i=2}^{n}\mathbb{Z}({\epsilon}_{i}-{\epsilon}_{i-1})\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}{L}_{p}=Q+\sum _{i=1}^{n}p\mathbb{Z}{\epsilon}_{i},& \text{(1.8)}\end{array}$$for each nonnegative integer $p\text{.}$ The lattice ${L}_{p}$ is a lattice of index $p$ in $L\text{.}$ Then

$${\mathcal{B}}_{\infty ,p,n}=\u27e8{X}^{\lambda},{T}_{i}\hspace{0.17em}\mid \hspace{0.17em}\lambda \in {L}_{p},2\le i\le n\u27e9$$and the group ${X}^{{L}_{p}}=\u27e8{x}^{\lambda}\hspace{0.17em}\mid \hspace{0.17em}\lambda \in {L}_{p}\u27e9$ is an abelian subgroup of ${\mathcal{B}}_{\infty ,p,n}\text{.}$

The affine Hecke algebra ${H}_{\infty ,1,n}$ (resp. ${H}_{\infty ,p,n}\text{)}$ is the quotient of the group algebra $\u2102{\mathcal{B}}_{\infty ,1,n}$ (resp. $\u2102{\mathcal{B}}_{\infty ,p,n}\text{)}$ by the relations

$${T}_{i}^{2}=(q-{q}^{-1}){T}_{i}+1,\phantom{\rule{2em}{0ex}}2\le i\le n\text{.}$$Let $L$ and ${X}^{\lambda}$ be as in (1.5) and (1.6). The subalgebra

$$\begin{array}{cc}\u2102\left[X\right]=\text{span}\{{X}^{\lambda}\hspace{0.17em}\mid \hspace{0.17em}\lambda \in L\}\phantom{\rule{1em}{0ex}}\text{(resp.}\hspace{0.17em}\u2102\left[{X}^{{L}_{p}}\right]=\text{span}\{{X}^{\lambda}\hspace{0.17em}\mid \hspace{0.17em}\lambda \in {L}_{p}\}\text{)}& \text{(1.10)}\end{array}$$is a commutative subalgebra of ${H}_{\infty ,1,n}$ (resp. ${H}_{\infty ,p,n}\text{).}$ The symmetric group ${S}_{n}$ acts on the lattice $L$ by permuting the ${\epsilon}_{i}$ and the lattices ${L}_{p}$ are ${S}_{n}\text{-invariant}$ sublattices of $L\text{.}$ Let ${s}_{i}=(i,i-1)\in {S}_{n}$ and ${\alpha}_{i}={\epsilon}_{i}-{\epsilon}_{i-1}\text{.}$ For $2\le i\le n$ and $\lambda \in L$ (resp $\lambda \in {L}_{p}\text{),}$

$$\begin{array}{cc}{X}^{\lambda}{T}_{i}={X}^{{s}_{i}\lambda}{T}_{i}+(q-{q}^{-1})\frac{{X}^{\lambda}-{X}^{{s}_{i}\lambda}}{1-{X}^{-{\alpha}_{i}}},& \text{(1.11)}\end{array}$$as elements of ${H}_{\infty ,1,n}$ (resp ${H}_{\infty ,p,n}\text{).}$

For each element $w\in {S}_{n}$ define ${T}_{w}={T}_{{i}_{1}}\dots {T}_{{i}_{p}}$ if $w={s}_{{i}_{1}}\dots {s}_{{i}_{p}}$ is an expression of $w$ as a product of simple reflections ${s}_{i}$ such that $p$ is minimal. The element ${T}_{w}$ does not depend on the choice of the reduced expression of $w$ [Bou1968, Ch. IV §2 Ex. 23]. The sets

$$\{{X}^{\lambda}{T}_{w}\hspace{0.17em}\mid \hspace{0.17em}\lambda \in L,w\in {S}_{n}\}\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}\{{X}^{\lambda}{T}_{w}\hspace{0.17em}\mid \hspace{0.17em}\lambda \in {L}_{p},w\in {S}_{n}\}$$are bases of ${H}_{\infty ,1,n}$ and ${H}_{\infty ,p,n},$ respectively [Lus1989]. The center of ${H}_{\infty ,1,n}$ is

$$\begin{array}{cc}Z\left({H}_{\infty ,1,n}\right)=\u2102{[{X}^{{\epsilon}_{1}},\dots ,{X}^{{\epsilon}_{n}}]}^{{S}_{n}}=\u2102{\left[X\right]}^{{S}_{n}},& \text{(1.12)}\end{array}$$and $\u2102{\left[{X}^{{L}_{p}}\right]}^{{S}_{n}}$ is the center of ${H}_{\infty ,p,n}$ (see Theorem 4.12 below).

This is an excerpt of the paper entitled *Affine Hecke Algebras, Cyclotomic Hecke algebras and Clifford Theory* authored by Arun Ram and Jacqui Ramagge. It was dedicated to Professor C.S. Seshadri on the occasion of his 70th birthday.

Research partially supported by the Natioanl Science Foundation (DMS-0097977) and the National Security Agency (MDA904-01-1-0032).